We investigate exact crossing minimization for graphs that differ from trees by a small number of additional edges, for several variants of the crossing minimization problem. In particular, we provide fixed parameter tractable algorithms for the 1-page book crossing number, the 2-page book crossing number, and the minimum number of crossed edges in 1-page and 2-page book drawings.

A straight-line drawing of a graph G is a mapping which assigns to each vertex a point in the plane and to each edge a straightline segment connecting the corresponding two points. The rectilinear crossing number of a graph G, cr(G), is the minimum number of pairs of crossing edges in any straight-line drawing of G. Determining or estimating cr(G) appears to be a difficult problem, and deciding...

The main purpose of this paper is to put recent results of Klazar and Noy [10], Kasraoui and Zeng [9], and Chen, Wu and Yan [2], on the enumeration of 2-crossings and 2-nestings in matchings, set partitions and linked partitions in the larger context of enumeration of increasing and decreasing chains in fillings of arrangements of cells. Our work is motivated by the recent paper of Krattenthale...

Zarankiewicz's Crossing Number Conjecture states that the crossing number cr(Km,n) of the complete bipartite graph Km,n equals Z(m, n) := m/2 (m − 1)/2n/2(n − 1)/2, for all positive integers m, n. This conjecture has only been verified for min{m, We determine, for each positive integer m, an integer N 0 = N 0 (m) with the following property: if cr(Km,n) = Z(m, n) for all n ≤ N 0 , then cr(Km,n)...

The aim of this paper is to endow a monoid structure on the set S of all oriented knots(links) under the operation ⊎ , called addition of knots. Moreover, we prove that there exists a homomorphism of monoids between (Sd, ⊎ ) to (N, +), where Sd is a subset of S with an extra condition and N is the monoid of non negative integers under usual addition.